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A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms
Abstract
The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.
... For a slant submanifold of a generalized Sasakian manifold the Maslov form is not always closed. At [5] we present a form that is always closed for a slant submanifold, so it really plays the role of the Maslov form in the cited papers. However, in the case where the submanifold satisfies the equality in the above inequality, such form is not always a conformal vector field. ...
This book contains an up-to-date survey and self-contained chapters on complex slant submanifolds and geometry, authored by internationally renowned researchers. The book discusses a wide range of topics, including slant surfaces, slant submersions, nearly Kaehler, locally conformal Kaehler, and quaternion Kaehler manifolds. It provides several classification results of minimal slant surfaces, quasi-minimal slant surfaces, slant surfaces with parallel mean curvature vector, pseudo-umbilical slant surfaces, and biharmonic and quasi biharmonic slant surfaces in Lorentzian complex space forms. Furthermore, this book includes new results on slant submanifolds of para-Hermitian manifolds.
... For a slant submanifold of a generalized Sasakian manifold the Maslov form is not always closed. In [5] we present a form that is always closed for a slant submanifold, so it really plays the role of the Maslov form in the cited papers. However, in the case where the submanifold satisfies the equality in the above inequality, such form is not always conformal. ...
The book gathers a wide range of topics such as warped product semi-slant submanifolds, slant submersions, semi-slant ξ^⊥-, hemi-slant ξ^⊥-Riemannian submersions, quasi hemi-slant submanifolds, slant submanifolds of metric f-manifolds, slant lightlike submanifolds, geometric inequalities for slant submanifolds, 3-slant submanifolds, and semi-slant submanifolds of almost paracontact manifolds. The book also includes interesting results on slant curves and magnetic curves, where the latter represents trajectories moving on a Riemannian manifold under the action of magnetic field. It presents detailed information on the most recent advances in the area, making it of much value to scientists, educators and graduate students.
Slant submanifolds were defined in the nineties by B.-Y. Chen and the corresponding theory has had an increasing development.2000 AMS Mathematics Subject Classification53C4053C4253C50
In this paper, we
show new results on slant submanifolds of an almost contact metric manifold. We
study and characterize slant submanifolds of K-contact and
Sasakian manifolds. We also study the special class of three-dimensional slant
submanifolds. We give several examples of slant
submanifolds.
In this paper, we study the possibility of obtaining an induced contact metric structure on a slant submanifold of a contact metric manifold. We also give a characterization theorem for three-dimensional slant submanifolds.
In this paper, we introduce two new classes of almost contact structures, called trans-Sasakian and almost trans-Sasakian structures, which are obtained from certain classes of almost Hermitian manifolds closely related to locally conformal Kahler or almost Kahler manifolds, respectively. In particular, although transSasakian structures are normal almost contact metric structures containing both cosymplectic and Sasakian structures, they are different from quasi-Sasakian structures, as it is shown constructing explicit examples, and in fact no inclusion relation between these classes exists.
We prove a simple geometric inequality for Lagrangian submanifolds of C^n. The equality case allows to characterized the Whitney immersion.
The present volume is the written version of a series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's recent work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.
We establish a canonical cohomology class on a proper slant submanifold of a Sasakian-space-form and give some geometrical obstructions to such immersions. Moreover, we combine both topo-logical and geometrical properties in order to show other interesting obstructions to slant immersions in contact geometry, involving the scalar curvature, the Ricci curvatures and a certain intrinsic function.
In the present paper submanifolds of generalized Sasakian-space-forms are studied. We focus on almost semi-invariant submanifolds, these generalize invariant, anti-invariant, and slant submanifolds. Sectional curva-tures, Ricci tensor and scalar curvature are also studied. The paper finishes with some results about totally umbilical submanifolds.
In this paper, contact metric and trans-Sasakian generalized Sasakian-space-forms are deeply studied. We present some general results for manifolds with dimension greater than or equal to 5, and we also pay a special attention to the 3-dimensional cases.
In this paper, we introduce the contact Whitney sphere as an imbedding of the n-dimensional unit sphere as an integral submanifold of the standard contact structure on R2n+1. We obtain a general inequality for integral submanifolds in R2n+1, involving both the scalar curvature and the mean curvature, and we use the equality case in order to characterize the contact Whitney sphere. We also study a similar problem for anti-invariant submanifolds of R2n+1, tangent to the structure vector field.